Definition:Totally Bounded Metric Space/Definition 1

From ProofWiki
Jump to navigation Jump to search

Definition

A metric space $M = \left({A, d}\right)$ is totally bounded if and only if:

for every $\epsilon \in \R_{>0}$ there exists a finite $\epsilon$-net for $M$.


That is, $M$ is totally bounded if and only if:

for every $\epsilon \in \R_{>0}$ there exists a finite set of points $x_1, \ldots, x_n \in A$ such that:
$\displaystyle A = \bigcup_{i \mathop = 1}^n B_\epsilon \left({x_i}\right)$
where $B_\epsilon \left({x_i}\right)$ denotes the open $\epsilon$-ball of $x_i$.


That is: $M$ is totally bounded if and only if, given any $\epsilon \in \R_{>0}$, one can find a finite number of open $\epsilon$-balls which cover $A$.


Also known as

A totally bounded metric space is also referred to as a precompact space.


Also see

  • Results about totally bounded metric spaces can be found here.


Sources